The property of the Kahan map in which we are interested in this paper is its preservation of the Poisson structure
of the associated Lotka-Volterra system, i.e., that the Kahan map, which is a birational automorphism of \(\mathbb {F}^n\) ,
is also a Poisson map (and hence a birational Poisson automorphism). Before giving the definition, let us clarify
the independence on \(\varepsilon \) : when the Kahan map is a Poisson map for some value of \(\varepsilon \)  then it is a
Poisson map for all values of \(\varepsilon \) . Indeed, the defining equations (REF ) of the Kahan map
are homogeneous when \(\varepsilon \)  is given weight \(-1\) , while giving a weight 1 to all \(x_i\) ; the claim then
follows from the fact that homotheties of quadratic Poisson structures are Poisson maps (see [1]}). It also shows that if the Kahan map of a Lotka-Volterra system \(\mathop {\rm LV}\nolimits (\Gamma )\)  is a Poisson map then
the Kahan map of the Lotka-Volterra system \(\mathop {\rm LV}\nolimits (\gamma \Gamma )\) , where \(\gamma \Gamma \)  is \(\Gamma \)  with all of its
values scaled by \(\gamma \in \mathbb {F}^*\) , is also a Poisson map. We therefore set \(\varepsilon =1\)  and when we speak of
the Kahan map or the Kahan morphism of a Lotka-Volterra system, it is implictly assumed that
\(\varepsilon =1\) . Notice that when \(\gamma =-1\) , then \(\gamma \Gamma =-\Gamma \)  is the graph \(\Gamma \)  with the direction
of all its arcs reversed.
